Subsurface geological modeling involves estimating parameters of interest in a 3-d model which is used for development planning and production forecasting. The modeling is a crucial part of the process of extraction of oil and gas from underground reservoirs. It is used, for example, initially to assess the viability of an underground reservoir for production and locate the sites for wells. It is also used to inform decisions about the rate of extraction of oil or gas which, if too rapid, can cause major technical problems. Decisions about the siting of future wells, after production has started, will also be informed by geomodelling.
A geological reservoir model should, as far as possible, be in conformity with the geological rules or features that are specific to the depositional environment of the reservoir of interest. The geomodel also, as far as possible, needs to be conditioned to all quantitative information from well cores and logs, seismic attributes, well tests and production history, etc. Thirdly, because of our incomplete knowledge about the subsurface reservoir, there is always a degree of uncertainty that needs to be accounted for in the reservoir modeling process.
In this context, several geostatistical or probabilistic methods have been developed for geological reservoir modeling. These methods allow the building of models that are representative of various depositional environments, with various types of data being integrated into the models. In addition, the nature of the probabilistic approach makes it possible to account for uncertainty.
Once a geostatistical reservoir model is chosen to describe the reservoir of interest, one can generate, potentially, an infinite number of so-called realizations. Each realization consists of a matrix comprising several values (representing parameters such as porosity and permeability) associated with each of a large number of cells distributed over the volume of the reservoir. Only for a relatively small number of these cells will the values be known with relative certainty (namely those cells which contain parameters which have been measured). For the remaining cells, the values are estimates based on the geostatistical modeling process. Each realization will have a different set of estimated values, each realization having been generated using a different, but equally valid, random seed.
The model is used to make predictions of e.g. flow rates and pressures in wells; these are sometimes known as the “dynamic responses” predicted by the model. Data (e.g. flow rate data) will be gathered from the wells over time once the reservoir is in production and, generally, these will differ from the predicted values generated by the model. History matching is a process by which new realizations of the model are generated which predict the correct current values for e.g. flow rate and pressure, and can therefore be assumed to be more accurate and to make more accurate predictions of these values for the future. In any history matching process creating updated realizations, a good goal is to preserve the geological features and statistical data on which the model is based.
In one history matching technique, an initial realization is modified so that the simulated (or predicted) dynamic responses are a better match with the measured ones. This procedure may be repeated independently for each of a number of initial realizations to build an averaged history matched realization. As mentioned above, for any modification, it is helpful if the statistical data and geological features inherent in the model are preserved.
In another history matching technique, the Ensemble Kalman filter (EnKF) is used for assimilating dynamic data. In the EnKF method, a series or “ensemble” of initial realizations is used, and they are simultaneously updated to dynamic data. This technique is normally applicable to continuous, Gaussian models.
The starting point for the EnKF process is a matrix comprising the ensemble of initial realizations. The dynamic response is computed for each initial realization, for a first time interval. The dynamic response could be as simple as a single flow rate number for each realization. However, it would more normally be a flow rate value and other data such as pressure and/or saturation, for each of a number of wells in the reservoir.
Once this calculation has been performed, a correlation matrix is constructed. This matrix contains the correlation coefficients between the dynamic responses and between the model parameters and dynamic responses. The data in the correlation matrix is sample covariance data which can be derived because a number of realizations are used instead of one.
The numbers of rows and columns of this matrix are respectively the number of dynamic responses and the sum of the number of model parameters and the number of dynamic responses. This is a large matrix and it takes a lot of computer resource for its construction, although it is nonetheless feasible on a current personal computer.
The next step is to combine the correlation matrix with the actual measured dynamic data from the reservoir over the first time interval, to produce an ensemble of updated realizations.
The process is iterated for the next time interval. Similarly, as new dynamic data from the reservoir become available, the EnKF process is applied again, starting with the latest updated ensemble of realizations from the most recent iteration of the EnKF process.
The details of the EnKF process itself are not the subject of the invention described in this patent application. However, a good explanation and overview of the EnKF process as it is applied in the present invention may be found in a helpful review paper, Aanonsen S I et al.: “The Ensemble Kalman Filter in Reservoir Engineering—A Review”, SPE Journal 14(3) (September 2009) 393-412. This paper is incorporated herein by reference.
A limitation of the EnKF method is that, strictly speaking, it applies only to continuous model parameters, and in particular to Gaussian parameters. It is efficient only if there is a linear statistical relationship between the flow response and the model parameters. Where the model is complex and discontinuous, the EnKF becomes less efficient. The limitations of the EnKF are discussed in the Aanonsen paper mentioned above (see, in particular, page 23 et seq.).
In practice, a facies model which is not continuous is often used. For some applications, a so-called truncated Gaussian model can be used to represent reservoirs with facies. In this model, the values of continuous Gaussian realizations are classified into intervals that represent facies so, for example, one facies may be represented by values between 1 and 2 in a realization and another facies represented by values between 0 and 1. It is possible to apply the EnKF on the continuous Gaussian realizations to perform history matching in such a discontinuous model.
For some complex facies structures, however, the truncated Gaussian model becomes unusable. Especially difficult are fluvial channel reservoirs which have very complex geological features. For reservoirs such as this, a different type of model is needed which represents facies in a different way. One such model is multiple point simulation (“MPS”). In MPS, the model is a function of the available facies data, a uniform random field and also a so-called training image, which is a conceptual representation of the facies geometry of the reservoir to be modelled.
The EnKF cannot be applied directly to an MPS model because it is discontinuous. Attempts have been made to overcome this problem. The so called “level set” and “discrete cosine transformation” methods seek to apply EnKF to MPS models. These methods are described in the Aanonsen S I et al paper referred to above. However, there is no mathematical proof that these techniques preserve the statistical data and geological features inherent in the model.